Let $G$ be a locally compact group. A continuous 
unitary representation $\pi$ of $G$ is said to be 
amenable if there exists a $G$-{\it invariant} mean 
on the space of $B(H_{\pi})$. In this paper, we 
investigate various characterization of amenable 
representations through the study of the existence 
and properties of invariant means on spaces of operators. 
One of our characterizations is an analogue of the {\it
Dixmier criterion} for amenable groups. We also attempt to
formulate a fixed point property for the amenable
representations.